Gregory Newton’s is a forward difference formula which is applied to calculate finite difference identity. Regarding the first value f0 and the power of the forward difference Δ, Gregory Newton’s forward formula gives an interpolated value between the tabulated points. The interpolated value is expressed by {fp}. By applying the forward difference operator and forward difference table, this method simplifies the calculations involved in the polynomial approximation of functions which are called spaced data points.
The Formula states for a ∈ [0,1]
Question 1
f(a) with the following data points,
| xi | 0 | 1 | 2 |
| fi | 1 | 7 | 23 |
Solution:
Forward difference table,
| xi | fi | Δfi | Δ2fi |
| 0 | 1 | ||
| 1 | 7 | 6 | |
| 2 | 23 | 16 | 10 |
a = 2
According to Gregory Newton’s forward difference formula,
f(0.5) = 1 + 2 × 6 + 2(2−1) x (10 / 2) + 2(2−1)x(2 – 2) x (6/6)
= 13 + 10 + 0
= 23
Therefore, f(2) = 23
Question 2
f(a) with the following data points,
| xi | 0 | 1 | 2 |
| fi | 1 | 7 | 23 |
Solution:
Forward difference table,
| xi | fi | Δfi | Δ2fi |
| 0 | 1 | ||
| 1 | 7 | 6 | |
| 2 | 23 | 16 | 10 |
a = 0.5
According to Gregory Newton’s forward difference formula,
f(0.5) = 1 + 0.5 × 6 + 0.5(0.5−1)×10 / 2 + 0.5(0.5−1)×(0.5 – 2)x6/6
= 1 + 3 + 2.5 × (-0.5) + (-0.25)(-1.5)
= (0.5)3 + 2(0.5)2 + 3(0.5) + 1
= 0.125 + 0.5 + 1.5 + 1
Therefore, f(0.5) = 3.125
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